Please check out the Changing Your Installation Directory instructions. This should hopefully help.
If you run into any problems, you can always create a fresh Showkase site on your new domain and then:
(1) Retain the Showkase settings from your original site by manually copying the following files and folders from your original site to your new site (overwriting the default versions which will be in place after a fresh install).
_data/sitedata
_data/themedata
_data/pagesprefs.txt
_data/theme.txt
Do not copy the _data/pagesdata.txt file though, as this lists the pages present in each site (which will be different after a fresh install).
Also, if you use a custom logo via an image stored in the library, for example, then you might also like to copy the '_library' folder from your original site to your new site.
(2) Import all the pages from your original site (via 'Site -> Import') to save having to recreate all your pages from scratch. You can import multiple pages at once so recreating your original site should be relatively quick and easy.
Hopefully the 'Move or Rename Installation' instructions will work fine for you but the procedure above should also work fine as a backup plan.